Paths, active paths, and d-separation in a finite DAG

A finite directed acyclic graph (DAG) is a finite vertex set $V$ with an arrow set $E\subseteq V\times V$ and no directed cycle. A path from $u$ to $v$ is a finite sequence of distinct vertices beginning at $u$ and ending at $v$ in which each consecutive pair is joined by an arrow in either direction; the arrows are not required to point along the sequence. An internal vertex of a path is a collider on that path when the two arrows on the path both point into it, and is a noncollider otherwise. A vertex $w$ is a descendant of $v$ if there is a directed path from $v$ to $w$; for this definition ${\mathrm{De}}^{+}(v)$ includes $v$ itself as well as its descendants.

For pairwise disjoint vertex sets $X,Y,C\subseteq V$, a path joining $X$ to $Y$ is active given $C$ when every internal noncollider is outside $C$ and every internal collider has at least one member of ${\mathrm{De}}^{+}(v)$ in $C$. The set $C$ d-separates $X$ and $Y$ in $G$ when no path joining $X$ to $Y$ is active given $C$; write $X{\perp }_{d}Y\mid C$ in $G$. The empty set of vertices is allowed for any of $X,Y,C$; in particular d-separation is vacuous if either endpoint set is empty.

For $T\subseteq V$, $G_{\underset{‾}{T}}$ is obtained by deleting every arrow whose tail lies in $T$, and $G_{\bar{T}}$ by deleting every arrow whose head lies in $T$. These are graph operations only; they do not by themselves specify a probability law under intervention.